10: polynomials © christine crisp “teach a level maths” vol. 1: as core modules
TRANSCRIPT
10: Polynomials10: Polynomials
© Christine Crisp
““Teach A Level Maths”Teach A Level Maths”
Vol. 1: AS Core Vol. 1: AS Core ModulesModules
Polynomials
Module C1
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Polynomials
Polynomials only contain terms of the type ,
where n is a positive integer
nax
The following are examples of Polynomial Functions:
Polynomial
Functions
10132 23 xxx
11324 xxx
32 xx A quadratic polynomial
A cubic polynomial
A quartic polynomial
Polynomials Expanding Cubic
Functions
e.g. 1
))()(( 125 xxxWe multiply 2 of the parts together first, leaving the third unchanged
))(( 1 x10x2 x52x
Polynomials
e.g. 1
))()(( 125 xxxWe multiply 2 of the parts together first, leaving the third unchanged
Now multiply each of the 3 terms in the 1st pair of brackets by each of the 2 terms in the 2nd
))(( 1 x10x2 x52x
))(( 11032 xxx
Expanding Cubic Functions
Polynomials
))(( 1 xxx 1032
e.g. 1
))()(( 125 xxxWe multiply 2 of the parts together first, leaving the third unchanged
Now multiply each of the 3 terms in the 1st pair of brackets by each of the 2 terms in the 2nd
xxx 103 23
))(( 1 x10x2 x52x
Expanding Cubic Functions
Polynomials
))((
e.g. 1
))()(( 125 xxxWe multiply 2 of the parts together first, leaving the third unchanged
Now multiply each of the 3 terms in the 1st pair of brackets by each of the 2 terms in the 2nd
xxx 103 23
))(( 1 x10x2 x52x
xxx 1032 1
10132 23 xxx
1032 xx
Expanding Cubic Functions
Polynomials
Expand the brackets in the following: )2)(1)(3( xxx
)2)(33( 2 xxxx
))(( 2322 xxx
642
322
23
xx
xxx
673 xx
Answer:
673 xx
Solution:
Exercise
)2)(1)(3( xxx
Polynomials Factorising Simple
Cubics
e.g. Factorise fully the following: xxxxf 54)( 23
Common factor:
Solution:
( Others are best done using the Factor Theorem which is covered later ).
Some cubic functions which contain a common factor can be factorised by inspection.
)54()( 2 xxxxf
Trinomial factors:
)1)(5()( xxxxf
We must now factorise the quadratic.
Polynomials
Factorise fully the following cubic:
xxx 82 23
)82( 2 xxx
)4)(2( xxx
Solution:
Exercise
xxx 82 23
Polynomials
Polynomials
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Polynomials
Polynomials only contain terms of the type ,
where n is a positive integer
nax
The following are examples of Polynomial Functions:
Polynomial
Functions
10132 23 xxx
11324 xxx
32 xx A quadratic polynomial
A cubic polynomial
A quartic polynomial
Polynomials
Expanding Cubic Functions
e.g. 1
))()(( 125 xxxWe multiply 2 of the parts together first, leaving the third unchanged
Now multiply each of the 3 terms in the 1st pair of brackets by each of the 2 terms in the 2nd
xxx 103 23
))(( 1 x10x2 x52x
)1)(103( 2 xxx
10132 23 xxx
1032 xx
Polynomials
Factorising Simple Cubics
e.g. Factorise fully the following: xxxxf 54)( 23
Common factor:
Solution:
( Others are best done using the Factor Theorem which is covered later ).
Some cubic functions which contain a common factor can be factorised by inspection.
)54()( 2 xxxxf
Trinomial factors:
)1)(5()( xxxxf
We must now factorise the quadratic.